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QPage 48-49: Q7: HC Verma: Concepts of Physics - Volume 1 Chapter-3 Rest and Motion Kinematics Page 48-49: Q7 Solution: Two particles A and B start from rest and move for equal time…

Two particles A and B start from rest and move for equal time on a straight line. The particle A has an acceleration a for the first half .....

Step-by-Step Verified Solution

Step 1: Define the variables

Let:

  1. Total time of motion = T
  2. For each particle, the motion is divided into two equal halves:
  3. First half: time = T/2
  4. Second half: time = T/2

Let a be some positive constant (acceleration).

Step 2: Particle A

First half (0 to T/2):

  1. Initial velocity: u₁ = 0
  2. Acceleration: a₁ = a
  3. Time: t₁ = T/2

Final velocity after first half:

v₁ = u₁ + a₁t₁ = 0 + a(T/2) = aT/2

Distance covered in first half:

s₁ = u₁t₁ + ½a₁t₁² = 0 + ½(a)(T/2)² = aT²/8

Second half (T/2 to T):

  1. Initial velocity: u₂ = v₁ = aT/2
  2. Acceleration: a₂ = 2a
  3. Time: t₂ = T/2

Distance covered in second half:

s₂ = u₂t₂ + ½a₂t₂²
s₂ = (aT/2)(T/2) + ½(2a)(T/2)²
s₂ = aT²/4 + a(T²/4)
s₂ = aT²/4 + aT²/4 = aT²/2

Total distance for Particle A:

S_A = s₁ + s₂ = aT²/8 + aT²/2
S_A = aT²/8 + 4aT²/8 = 5aT²/8

Step 3: Particle B

First half (0 to T/2):

  1. Initial velocity: u₁ = 0
  2. Acceleration: a₁ = 2a
  3. Time: t₁ = T/2

Final velocity after first half:

v₁ = u₁ + a₁t₁ = 0 + (2a)(T/2) = aT

Distance covered in first half:

s₁ = u₁t₁ + ½a₁t₁² = 0 + ½(2a)(T/2)²
s₁ = a(T²/4) = aT²/4

Second half (T/2 to T):

  1. Initial velocity: u₂ = v₁ = aT
  2. Acceleration: a₂ = a
  3. Time: t₂ = T/2

Distance covered in second half:

s₂ = u₂t₂ + ½a₂t₂²
s₂ = (aT)(T/2) + ½(a)(T/2)²
s₂ = aT²/2 + aT²/8
s₂ = 4aT²/8 + aT²/8 = 5aT²/8

Total distance for Particle B:

S_B = s₁ + s₂ = aT²/4 + 5aT²/8
S_B = 2aT²/8 + 5aT²/8 = 7aT²/8

Step 4: Compare


Particle

Total Distance

A

5aT²/8

B

7aT²/8

Clearly:

S_B > S_A

Step 5: Intuitive Explanation

Why does Particle B cover more distance even though both use the same two accelerations (a and 2a) for equal times?

  1. Particle B gets the larger acceleration (2a) first, when it is starting from rest.
  2. This gives it a higher velocity early on.
  3. Even though its second-half acceleration is smaller (a), it enters the second half with a much larger initial speed (aT vs. aT/2 for Particle A).
  4. Since distance depends heavily on initial velocity (especially in the second half), Particle B maintains a significant lead.

In contrast, Particle A gets the smaller acceleration first, so it is slower during the entire first half and never catches up.

Final Answer

Particle B covers the larger distance.


Particle

Total Distance

A

5aT²/8

B

7aT²/8

So, B > A by a margin of aT²/4.

All Questions in this Chapter

QP 49-50: Q1 QP 49-50: Q2 QP 49-50: Q3 QP 49-50: Q4 QP 49-50: Q5 QP 49-50: Q6 QP 49-50: Q7 QP 49-50: Q8 QP 49-50: Q9 QP 49-50: Q10 QP 49-50: Q11 QP 49-50: Q12 QP 49-50: Q13 QP 50-51: Q1 QP 50-51: Q2 QP 50-51: Q3 QP 50-51: Q4 QP 50-51: Q5 QP 50-51: Q6 QP 50-51: Q7 QP 50-51: Q8 QP 50-51: Q9 QP 50-51: Q10 QP 51-54: Q1 QP 51-54: Q2 QP 51-54: Q3.1 QP 51-54: Q3.2 QP 51-54: Q3.3 QP 51-54: Q3.4 QP 51-54: Q4.1 QP 51-54: Q4.2 QP 51-54: Q5 QP 51-54: Q6 QP 51-54: Q7 QP 51-54: Q8.1 QP 51-54: Q8.2 QP 51-54: Q8.3 QP 51-54: Q9.1 QP 51-54: Q9.2 QP 51-54: Q10 QP 51-54: Q11 QP 51-54: Q12 QP 51-54: Q13 QP 51-54: Q14 QP 51-54: Q15.1 QP 51-54: Q15.2 QP 51-54: Q15.3 QP 51-54: Q16 QP 51-54: Q17 QP 51-54: Q18.1 QP 51-54: Q18.2 QP 51-54: Q19 QP 51-54: Q20 QP 51-54: Q21 QP 51-54: Q22 QP 51-54: Q23.1 QP 51-54: Q23.2 QP 51-54: Q23.3 QP 51-54: Q24 QP 51-54: Q25.1 QP 51-54: Q25.2 QP 51-54: Q25.3 QP 51-54: Q26 QP 51-54: Q27 QP 51-54: Q28 QP 51-54: Q29 QP 51-54: Q30 QP 51-54: Q31 QP 51-54: Q32.1 QP 51-54: Q32.2 QP 51-54: Q32.3 QP 51-54: Q33.1 QP 51-54: Q33.2 QP 51-54: Q34 QP 51-54: Q35 QP 51-54: Q36 QP 51-54: Q37 QP 51-54: Q38 QP 51-54: Q39 QP 51-54: Q40 QP 51-54: Q41 QP 51-54: Q42 QP 51-54: Q43.1 QP 51-54: Q43.2 QP 51-54: Q44 QP 51-54: Q45 QP 51-54: Q46.1 QP 51-54: Q46.2 QP 51-54: Q47.1 QP 51-54: Q47.2 QP 51-54: Q48 QP 51-54: Q49.1 QP 51-54: Q49.2 QP 51-54: Q50 QP 51-54: Q51 QP 51-54: Q52 QPage 48-49: Q1 QPage 48-49: Q2 QPage 48-49: Q3 QPage 48-49: Q4 QPage 48-49: Q5 QPage 48-49: Q6 QPage 48-49: Q7 QPage 48-49: Q8 QPage 48-49: Q9 QPage 48-49: Q10.1 QPage 48-49: Q10.2 QPage 48-49: Q10.3 QPage 48-49: Q11 QPage 48-49: Q12 QPage 48-49: Q13 QPage 48-49: Q14

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